Question 8 complete, please solve briefly stating every reason, its from subject Number theory lgebra Fc*Stephen H. Friedberg, ArnoldCBCS BSc(Hons)Maths (3).pdfPL Elementary_Number_Theory 7th XPDEle C:/Users/Dell/Downloads/SEM-6%20%20BOOKS/Elementary Number Theory 7th edition%20(1).pdfMaps|| 三业三业三三三|=ル三: | 三 | 三 =三三川三业三川彡山NewsTranslate|| 三 |三: 小三小三业业业三一 1三心彡W彡1彡彡山三业三 |三 |三山三川三ll=山三+.1彡业| 三业三1三 = 11 业=8k + 1.[Hint: Assume that there are only finitely many primes of the form 4k + 1; call themPi, P2, . . . , Pr. Consider the integer (2p1 p2 Pr)² +1 and apply the previous prob-lem.]8. (a) Prove that if p and q are odd primes and q |a' = 1, then either q|a–1 or elseq = 2kp + 1 for some integer k.[Hint: Because aP = 1 (mod q), the order of a modulo q is either 1 or p; in the lattercase, p| ø(q).](b) Use part (a) to show that if p is an odd prime, then the prime divisors of 2P – 1 areof the form 2kp + 1.(c) Find the smallest prime divisors of the integers 217= 1 and 229 – 1.9. (a) Verify that 2 is a primitive root of 19, but not of 17.(b) Show that 15 has no primitive root by calculating the orders of 2, 4, 7, 8, 11, 13, and14 modulo 15.10. Let r be a primitive root of the integer n. Prove that r is a primitive root of n if and onlyif gcd(k, o(n)) = 1.11. (a) Find two primitive roots of 10.(b) Use the information that 3 is a primitive root of 17 to obtain the eight primitive rootsof 17.12. (a) Prove that if p and a 3 are both odd primes and a | Rn, then a = 2kp + 1 for someDE) 11°CMovies & TVO Elementary_Numb..CapturesarchDELLInsertDeleteF12PriScrF10F11F8F9F6Beck5 €R

Question

Question 8 complete, please solve briefly stating every reason, its from subject Number theory lgebra Fc*Stephen H. Friedberg, ArnoldCBCS BSc(Hons)Maths (3).pdfPL Elementary_Number_Theory 7th XPDEle C:/Users/Dell/Downloads/SEM-6%20%20BOOKS/Elementary Number Theory 7th edition%20(1).pdfMaps|| 三 业三业三 三 三|=ル三: | 三 | 三 =三 三川三业三川彡山NewsTranslate|| 三 |三: 小三小三业 业 业三一 1三心彡W彡1彡 彡山三业三 |三 |三 山三川三ll=山三+.1彡业| 三 业三1三 = 11 业=8k + 1.[Hint: Assume that there are only finitely many primes of the form 4k + 1; call themPi, P2, . . . , Pr. Consider the integer (2p1 p2 Pr)² +1 and apply the previous prob-lem.]8. (a) Prove that if p and q are odd primes and q |a' = 1, then either q|a–1 or elseq = 2kp + 1 for some integer k.[Hint: Because aP = 1 (mod q), the order of a modulo q is either 1 or p; in the lattercase, p| ø(q).](b) Use part (a) to show that if p is an odd prime, then the prime divisors of 2P – 1 areof the form 2kp + 1.(c) Find the smallest prime divisors of the integers 217= 1 and 229 – 1.9. (a) Verify that 2 is a primitive root of 19, but not of 17.(b) Show that 15 has no primitive root by calculating the orders of 2, 4, 7, 8, 11, 13, and14 modulo 15.10. Let r be a primitive root of the integer n. Prove that r is a primitive root of n if and onlyif gcd(k, o(n)) = 1.11. (a) Find two primitive roots of 10.(b) Use the information that 3 is a primitive root of 17 to obtain the eight primitive rootsof 17.12. (a) Prove that if p and a > 3 are both odd primes and a | Rn, then a = 2kp + 1 for someDE) 11°CMovies & TVO Elementary_Numb..CapturesarchDELLInsertDeleteF12PriScrF10F11F8F9F6Beck5 €R

Accepted Answer

a)
Given that p and q are odd primes and qap-1.
So,
ap≡1mod q
If p=1 then 
a≡1mod q
So, qa-1
If p≠1…